Remote Sensing of Water and Environment
Chapter 7: Microwave Radiometry and Radiative Transfer
Ardeshir Ebtehaj
University of Minnesota
Table of Contents
Chapter 7: Microwave Radiometry and Radiative Transfer
7-1- Planck's Blackbody Radiation Law
7-2- The Rayleigh-Jeans Law
7-3- Antenna Pattern
7-4- Power and Temperature Correspondence
7-5- Radiation by Natural Materials
7-5-1- Brightness Temperature
7-6- Distribution of Brightness Temperature
7-6- Theory of Radiative Transfer
7-6-1 Equation of Radiative Transfer
7-6-2 Theory of Radiative Transfer for Brightness Temperature
7-6-3 Brightness Temperature of a Stratified Medium
7-6-4 Brightness Temperature of a Scatter-Free Medium
7-6-5 Upwelling and Downwelling Atmospheric
7-7- Terrain Brightness Temperature
7-7-1 Transmission Across a Specular Boundary
7-7-2 Emission by a Specular Surface
7-7-3 Emission by a Rough Surface
The bulk of the energy recevied by the Earth is in the form of solar EM radiation. Part of this incident radiation is scattered or absorbed by the Earth's atmosphere and the remainder is transmitted to the Earth's surface. Of the EM waves that reach the surface, some will be scattered outward and the rest will be absorbed. Absorbed EM energy is transformed into heat.
The transformation process of EM energy across the Earth's materials is explained by the theory of radiative transfer.
Radiometry is the field of science and engineering concerned with the measurement of EM radiation.
The term radiometry means the measurement of incoherent EM energy. All materials emit (radiate) EM energy.
- All substances at a finite absolute temperature radiate EM energy.
- Atomic gases radiate EM waves at discrete frequencies, or wavelengths. In other words, they have line spectra (e.g. oxygen absorbs/emits @ 60 GHz).
- The intensity of energy radiated by a substance increases with its temperature.
- Liquid and solid materials have a continuous spectrum of radiation.
7-1- Planck's Blackbody Radiation Law
A blackbody radiates uniformly in all directions with a spectral brightness intensity, 
where
- h: Planck's constant (
) - f: frequency
- κ: Boltzmann's constant (
) - T: absolute temperature [K]
- c: velocity of light (
)
Over a narrow frequency interval centered at f, the brightness intensity is given by
It is sometimes of interest to express the brightness intensity in terms of wavelength λ. Knowing that 
Planck's radiation law [adopted from Kraus, 1966]
Geometry for power received from a blackbody source
Let's asume that is the power [watts] emitted by a blackbody source area of 
, through a solid angle of 
, and over a bandwidth of 
. The blackbody source has an area of 
. Our objective is to determine spectral power 
intercepted by a receiver with an aperture area at 
at a range R from the source.
is the power [watts] emitted by a blackbody source area of , through a solid angle of , and over a bandwidth of . The blackbody source has an area of . Our objective is to determine spectral power intercepted by a receiver with an aperture area at at a range R from the source. 
[
]
And thus,
For a receiver with a bandwidth 
.7-2- The Rayleigh-Jeans Law
The dashed line in the following figure provides an excellent approximation to Planck's law, as long as the frequency f is well below 
, where is the maximum. The law that explains this dashed line is known as the Rayleigh-Jeans law.
, where is the maximum. The law that explains this dashed line is known as the Rayleigh-Jeans law. 
Figure 2-1: Comparison of Planck's law with its low-frequency approximation (Rayleigh-Jeans law) at 300 K.
Note: 
Thus, if 
in Planck's law, the expression simplifies to
in Planck's law, the expression simplifies to 
Eq. (7.1)The law is valid when 
or 
. Above 300 GHz, the deviation from the Planck's law is around 3%.
or . Above 300 GHz, the deviation from the Planck's law is around 3%. 7-3- Antenna Pattern
Each specific combination of the zenith angle 
and the azimuth angle 
denotes a specific direction in a spherical coordinate system.
and the azimuth angle denotes a specific direction in a spherical coordinate system. The directional pattern of an antenna is described by its normalized radiation intensity 
, defined as the ratio of the power density 
at a specified range R to 
, the maximum value of at the same range,
, defined as the ratio of the power density at a specified range R to , the maximum value of at the same range,
(dimensionaless).The normalized radiation intensity 
characterizes the directional pattern of the energy radiated or recieved by the antenna.
characterizes the directional pattern of the energy radiated or recieved by the antenna. Some antennas exhibit highly directive patterns with narrow beams, in which case it is often convenient to plot the antenna pattern on a decibel scale by expressing in decibels:
in decibels:
Blackbody spectral brightness incident antenna with effective aperture and radiation pattern .
incident antenna with effective aperture and radiation pattern .
Representative plots of the normalized radiation pattern of a microwave antenna in (left) polar form and (right) rectangular form.
For an antenna with a single mainlobe, the pattern solid angle 
describes the equivalent width of the mainlobe of the antenna pattern as shown in the following figure.
describes the equivalent width of the mainlobe of the antenna pattern as shown in the following figure.
The pattern solid angle defines an equivalent cone over which all the radiation of the actual antenna is concentrated with uniform intensity equal to the maximum of the actual pattern.
defines an equivalent cone over which all the radiation of the actual antenna is concentrated with uniform intensity equal to the maximum of the actual pattern.7-4- Power and Temperature Correspondence
Consider the above arrangement for a lossless receiver antenna with effective aperture , surrounded by a blackbody of spectral intensity . The antenna is characterized by a radiation pattern . Therefore
, surrounded by a blackbody of spectral intensity . The antenna is characterized by a radiation pattern . Therefore
The total power received is
Emission by a blackbody is unpolarized. Therefore, assume half of the energy is in vertical and half is in horizontal polarization.
Using the Rayleigh-Jeans approximation, we have
If the detected power is limited to a narrow bandwidth, 
.
.
Solving the integral using:
,leads to the final equation
This result is of fundamental significance in microwave remote sensing. The direct linear relationship between power and temperature lends to the interchangeable use of them. Note thet 
is radiation power by a balck body.
is radiation power by a balck body.7-5- Radiation by Natural Materials
7-5-1- Brightness Temperature
A blackbody is an idealized body and a perfect absorber.
Real materials, sometimes referred to as gray body, emit less than a blackbody and do not necessarily absorb all the incident energy upon them.
In the microwave region, given Eq. (6.2.1), the unpolarized brightness intensity 
of a blackbody at temperature T is
of a blackbody at temperature T is
For a gray body, the directional emission/absorption, where the Rayleigh-Jeans law is valid in microwave bands, is as follows:
where 
is the brightness temperature [K]. The emissivity, 
is expressed as
is the brightness temperature [K]. The emissivity, is expressed as---------------------------------------------------------IMPORTANT
Brightness temperature of a semi-infinite isothermal medium.
Since 
, it follows that 
. Thus, 
. It is customary to use instead of 
, especially in microwave region.
, it follows that . Thus, . It is customary to use instead of , especially in microwave region. ---------------------------------------------------------IMPORTANT
7-6- Distribution of Brightness Temperature
The radiation incident upon the antena from any specific direction may contain components originating from several different sources.
- Upward surface emission -
- Upward atmospheric emission -
- Downward atmospheric emission -
that gets scattered/reflected by the terrain (surface scattering) in the direction of antenna 
Relationship between (lossless) antenna temperature 
, brightness temperature incident on the antenna and surface emitted brightness temperature 
: a) schematic representation; b) block-diagram representation.
, brightness temperature incident on the antenna and surface emitted brightness temperature : a) schematic representation; b) block-diagram representation.The net brightness temperature 
, representing the energy incident upon the antenna, is given by
, representing the energy incident upon the antenna, is given by
Examples of configurations of interest in radiometric remote sensing.
7-6- Theory of Radiative Transfer
The purpose here is to develop formulations to relate observed brightness temperatures incident upon radiometer antenna aperture to the physical and electromagnetic properties of the scene.
incident upon radiometer antenna aperture to the physical and electromagnetic properties of the scene. Interaction between electromagnetic radiation and matters involves two simultaneous processes:
- Extinction: radiation tranversing a medium is reduced in intenity due to absorption and/or scattering by the medium.
- Emission: when the medium adds energy of its own to the original beam of energy.
Radiation transfer through an infinitesimal cylinder
7-6-1 Equation of Radiative Transfer
: incident normally upon the lower face of the cylinder- R: location of the cylinder
: direction of the cylinder (brightness intensity)
Brightness intensity is energy per unit area dA, radiated within a solid angle 
, during an interval of 
with a frequency bandwidth . The loss in intensity due to propogation over thickness dR is
, during an interval of with a frequency bandwidth . The loss in intensity due to propogation over thickness dR is 
(Eq. 7.2)where 
is the extinction coefficient 
. The extinction coefficient can be expressed as
is the extinction coefficient . The extinction coefficient can be expressed as 
where 
and 
are obsorption and scattering coefficients.
and are obsorption and scattering coefficients.In addition to alternating the incident brightness temperature, the medium adds its own emission along :
:
where 
and 
are source functions:
and are source functions:- : absorption source function
- : scattering source function
The scattering source function 
accounts for radiation scattered in the direction
due to radiation incident upon the cylinder from all other directions.
accounts for radiation scattered in the direction due to radiation incident upon the cylinder from all other directions. 
where 
is the brightness intensity incident from the direction 
, a portion of which gets redirected along the direction , thereby adding to the original brightness intensity.
is the brightness intensity incident from the direction , a portion of which gets redirected along the direction , thereby adding to the original brightness intensity. Scattering is not necessarily an isotropic process due to the shape, size, orientation and spatial distribution of particles that scatter. This heterogeneity is accounted for by 
, or the scattering phase function. In an isotropic medium,
, or the scattering phase function. In an isotropic medium,
Oftentimes, and 
are expressed as
and are expressed as -----------------------------------------------------------------------------------------------
(single scattering albedo) 
-----------------------------------------------------------------------------------------------
(Eq. 7.3)where 
is called a total source function. Propogation over thickness dR results in a change in intensity dI given by
is called a total source function. Propogation over thickness dR results in a change in intensity dI given by
The dimensionless 
is called the optical or electromagnetic depth.
is called the optical or electromagnetic depth. ----------------------------------------------------------------------------------------------------------------------------------
Therefore, the radiative tranfer equation is
----------------------------------------------------------------------------------------------------------------------------------
7-6-2 Theory of Radiative Transfer for Brightness Temperature
Rayleigh-Jeans law:
Absorption source function:
- As noted earlier, Kirchhoff’s law states that under conditions of local thermodynamic equilibrium, thermal emission has to be equal to absorption, which leads to the conclusion that the absorption source function
is isotropic and given by the Rayleigh–Jeans approximationto Planck’s law and thus independent of 
.
Scattering source function:
where 
is the volume scattering radiometric temperature.
is the volume scattering radiometric temperature.
Therefore, given that , the radiative transfer equation for brightness temperature is
, the radiative transfer equation for brightness temperature is--------------------------------------------------------------------------------------------------------
Eq. (7.4)--------------------------------------------------------------------------------------------------------
7-6-3 Brightness Temperature of a Stratified Medium
Geometry for brightness temperature radiative transfer equation
Let us assume that propogation properties only change as a function of z. We want to solve the radiative transfer equation to obtain 
.
.
Let us multiple both sides of the radiative transfer equation for , Eq. (7.4.), by 
, Eq. (7.4.), by 
where 
. The left handed side can be written as
. The left handed side can be written as
(Eq. 7.5)In the last step we used for 
which can be proven by
----------------------------------------------------------------------------------
At a height 
, Eq. (7.5) becomes
, Eq. (7.5) becomes 
where 
is the upward emission contribution
is the upward emission contribution
(Eq. 7.6)and 
is the surface emission + reflected downward atmospheric or vegetation emission.
is the surface emission + reflected downward atmospheric or vegetation emission. ----------------------------------------------------------------------------------
Note that gets attenuated from 
to 
along its traveling path. The one-way atomspheric transmissivity between 
and is
gets attenuated from to along its traveling path. The one-way atomspheric transmissivity between and is
.- The frist term in the right hand side of (Eq. 7.6) is the emitted radiation attenuated by
from 
to H is
- The second term is the energy scattered into the differential thickness attentuated by from to H is
.7-6-4 Brightness Temperature of a Scatter-Free Medium
In a scatter free medium we can assume 
. Therfore, Eq. (7.5) reduces to
. Therfore, Eq. (7.5) reduces to 
(optical depth)For a constant 
and temperature, we can reduce the above equation to the following
and temperature, we can reduce the above equation to the following 
and 
- Under clear-sky conditions,
(not in cloudy conditions) - Scattering can be ignored for most atmospheric conditions (
) - Absorption is generated by an average conductivity of the medium, while scattering is controlled by the spatial heterogeneity of the medium relative permittivity
- Scattering by particles inside of a volume is called volume scattering, which should not be confused with surface scattering
Volume Scattering
Snow particles range from 0.1-5 mm. If the wavelength is much larger than the grain size, the medium appead electromagnetically homogeneous with no appreciable scattering. If the wavelength is on the same order of magnitude as particle size, scattering will take place.
Scatter-free
Scattering will take place7-6-5 Upwelling and Downwelling Atmospheric 
In the absence of scattering, the upwelling atmospheric brightness temperature is
.Models for 
and 
are available for Earth's atmosphere.
and are available for Earth's atmosphere. 
Upward and downward emission contributions of a plane-stratified atmosphere
A ground-based upward-looking radiometer observes a downwelling atmospheric brightness temperature given by
The physical interpretation of the above equation is quite simple; the energy emitted by a stratum at height and of vertical thickness 
(slant thickness 
) is proportional to 
, which, after propagation down to the surface, is reduced by the factor 
due to absorption by the intervening layers.
and of vertical thickness (slant thickness ) is proportional to , which, after propagation down to the surface, is reduced by the factor due to absorption by the intervening layers.For a planar, homogeneous atmosphere with 
, 
, and depth H,
, , and depth H,
Note that where 
is the zenith optical thickness of the atmospheric layer
is the zenith optical thickness of the atmospheric layer
so for a very optically thick atmosphere 
, Then we have
, Then we have
7-7- Terrain Brightness Temperature
Three scenarios for medium 2 and its surface boundary.
7-7-1 Transmission Across a Specular Boundary
Brightness transmission across a planar boundary
The power of the EM wave, for example in horizontal polarization, is
Base on the the Snell's Law, we have
Differentiating the law, one can have
Multiplying both sides with the azimuth angle, we have
Multiplying both sides of the above equation by the Snell's law, one can get
we know that 
and thus the above equation simplifies to
and thus the above equation simplifies to
Given that
, 
, 
where 
are the brightness intensities for a grey body medium. Therefore,
are the brightness intensities for a grey body medium. Therefore, 
or
where the trasnissivity when EM waves travel from medium 2 to medium 1 is
, in which 
are the Fresnel reflectivity for incidence angles.
, in which are the Fresnel reflectivity for incidence angles. Similarly for the vertical polarization, we have
where
7-7-2 Emission by a Specular Surface
Medium 2 has a temperature-depth profile 
We already described that the slanted emission by a medium can be expleained by the following equation.
: Temperature profile medium 2- : Absorption coefficient
For a short interval over which 
is constant
is constant 
(note that 
).We have shown that 
.
.If and constitutive parameters of medium 2 are known, we can solve the above equation numerically. For a homogeneous medium with a uniform temperautre 
, the emission simplifies to 
.
and constitutive parameters of medium 2 are known, we can solve the above equation numerically. For a homogeneous medium with a uniform temperautre , the emission simplifies to . Therefore, observed brightness temperature by a downlooking radiometer is
where we know that 
.
. Note: 
is temperature of uppermost layer that represents bulk of the contributions to 
.
is temperature of uppermost layer that represents bulk of the contributions to .
Calculated reflectiveness and emissivities of a specular surface for h and v polarizations at 10 GHz
7-7-3 Emission by a Rough Surface
Specular- and rough-surface sacattering and emission
Generally speaking, increasing surface roughness increases surface emssion radiation than that of a specular surface with the same dielectic constant.
There are semi empirical models for explaining total reflectivity 
. The basic structure of 
,
. The basic structure of , where 
is an effective rough surface reflectivity given by
is an effective rough surface reflectivity given by
where 
and 
are Fresnel reflectivity at polarization p, Q is a polarization mixing factor, 
is an equivalent root mean squared (rms) height and n is an angular exponent, which is generally polarization dependent.
and are Fresnel reflectivity at polarization p, Q is a polarization mixing factor, is an equivalent root mean squared (rms) height and n is an angular exponent, which is generally polarization dependent.Parameters derived from regression analyses (Wang et al., 1983)
where 
: electromagnetic roughness
: electromagnetic roughness
Angular patterns of the emissivity measured at 1.4 GHz for three bare-soil fields with different surface roughness [Newton and Rouse, 1980]
Two configurations of height variations
For a random surface whose mean is coincide with x-y plane, its height 
above x-y plane usually is characterized by a Gaussian density 
:
above x-y plane usually is characterized by a Gaussian density :
, where s is the rms heightwhere 
and 
.
and .The correlation function 
is defined as
is defined as
.The correlation length l is the value of ξ at which 
.
.IMPORTANT: A surface is radiometrically smooth when 
.
.
Random, isotropic surface : (a) pictorial view, (b) measured height profile 
, (c) pdf of digitized height profile, and (d) autocorrelation function 
where ζ is the displacement between two points on the surface
: (a) pictorial view, (b) measured height profile , (c) pdf of digitized height profile, and (d) autocorrelation function where ζ is the displacement between two points on the surface